Analysis of Change

Authors

Affiliations

Nate Yomogida, B.S., SPT

Doctor of Physical Therapy

B.S. in Kinesiology

Chloë Kerstein, B.S., SPT

Doctor of Physical Therapy

B.A. in Neuroscience

Calvin & Hobbes - Bill Waterson¹

Absolute Change

Absolute change (Δy) is the simplest metric with which to assess change².

\[ \Delta y = y_f - y_i \]

Problems

The absolute change (Δ y) may depend, in part, on the initial value \(y_i\)².

Variation between groups

• If \(y_i\) varies substantially between groups suppose \(y_i\) represents the initial diameter of arteries and arterioles then larger values of Δy are likely to be associated with larger values of \(y_i\)².

Variation within groups

• Oppositely, if yi varies within a group
• Example: suppose now that \(y_i\) represents the initial blood pressure of healthy controls
• For mathematical reasons alone, if \(y_i\) is smaller, then \(y_f\) can increase more:²
  • The lower you start, the higher you can climb.
• If \(y_i\) is bigger, then \(y_f\) can decrease more:
  • the higher you start, the farther you can fall².

Solution

Standardized Absolute change

\[ \textrm{Standardized Absolute Change} = \frac{\Delta y}{y_i} = \frac{y_f - y_i}{y_i} \]

Note

Often, this ratio is rescaled to percent change (%Δ), by multiplying by 100²

Percent Change (%Δ)

\[ \textrm{Percent Change} = \textrm{Standardized Absolute Change} \times 100\% = \frac{\Delta y}{y_i} \times 100\% \]

Problems

Undefined Results

If the initial value happens to be 0, then the percent change is undefined².

Infinity

And, as yi gets smaller and smaller, %Δ gets bigger and bigger, approaching \(\infty\) or \(1 - \infty\) depending on whether \(y_f\) increases or decreases from \(y_i\)².

Direction of comparison guides magnitude

Increasing

• \(y_i = 1\)
• \(y_f = 2\)
• \(\frac{\Delta y}{y_i} \times 100 \% = \frac{2-1}{1} \times 100 \% = \textbf{100\%}\)
• We can conclude that \(y_f\) is 100% greater than \(y_i\)²

Decreasing

• \(y_i = 2\)
• \(y_f = 1\)
• \(\frac{\Delta y}{y_i} \times 100 \% = \frac{1 - 2}{2} \times 100 \% = \textbf{50\%}\)
• We can conclude that the \(y_f\) is 50% less than \(y_i\).

We can resolve this logical dissonance using Symmetrized Percent change

²

Symmetrized Percent Change

Symmetrized %Δ is better behaved mathematically than is %Δ².

\[ \textrm{Symmetrized Percent Change} = \frac{y_f - y_i}{y_f + y_i} \times 100\% = \frac{\Delta y}{y_f + y_i} \times 100\% \]

• If \(y_i\) happens to be 0, then the symmetrized %Δ is 100%².
• If \(y_i\) differs from 0 and \(y_f = 0\), then the symmetrized %Δ is \(-100\%\)².
• As with %Δ,if \(y_f = y_i\), then the symmetrized %Δ is 0%². And a pleasing property (24) is that, for some pair of initial and final values, the magnitude of the symmetrized percent change is unaffected by the direction of the comparison².

\[ \frac{y_f - y_i}{y_f + y_i} \times 100\% = \frac{2-1}{1+2} \times 100\% = 33\% \]

\[ \frac{y_f - y_i}{y_f + y_i} \times 100\% = \frac{1-2}{2+1} \times 100\% \]

Percent Change vs Symmetrized Percent Change

• These two metrics of relative change purport to account for differences in the initial value².
• Similar to other ratios, if there is no relationship between Δy and \(y_i\), then the mere calculation of %Δ and symmetrized %Δ creates a relationship². If there is a relationship between Δy and \(y_i\), then the calculation of %Δ and symmetrized %Δ exaggerates the strength of that relationship².

Which test to use?

Metrics of change plotted against initial value²

References

1.

Watterson B. The Complete Calvin and Hobbes. 1st ed. Andrews McMeel Publishing, LLC; 2012.

2.

Curran-Everett D, Williams CL. Explorations in statistics: The analysis of change. Advances in Physiology Education. 2015;39(2):49-54. doi:10.1152/advan.00018.2015